1,894 research outputs found
Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions
Three-dimensional integer partitions provide a convenient representation of
codimension-one three-dimensional random rhombus tilings. Calculating the
entropy for such a model is a notoriously difficult problem. We apply
transition matrix Monte Carlo simulations to evaluate their entropy with high
precision. We consider both free- and fixed-boundary tilings. Our results
suggest that the ratio of free- and fixed-boundary entropies is
, and can be interpreted as the ratio of the
volumes of two simple, nested, polyhedra. This finding supports a conjecture by
Linde, Moore and Nordahl concerning the ``arctic octahedron phenomenon'' in
three-dimensional random tilings
Two-dimensional random tilings of large codimension: new progress
Two-dimensional random tilings of rhombi can be seen as projections of
two-dimensional membranes embedded in hypercubic lattices of higher dimensional
spaces. Here, we consider tilings projected from a -dimensional space. We
study the limiting case, when the quantity , and therefore the number of
different species of tiles, become large. We had previously demonstrated [ICQ6]
that, in this limit, the thermodynamic properties of the tiling become
independent of the boundary conditions. The exact value of the limiting entropy
and finite corrections remain open questions. Here, we develop a mean-field
theory, which uses an iterative description of the tilings based on an analogy
with avoiding oriented walks on a random tiling. We compare the quantities
so-obtained with numerical calculations. We also discuss the role of spatial
correlations.Comment: Proceedings of the 7th International Conference on Quasicrystals
(ICQ7, Stuttgart), 4 pages, 4 figure
Entanglement entropy of fermions in any dimension and the Widom conjecture
We show that entanglement entropy of free fermions scales faster then area
law, as opposed to the scaling for the harmonic lattice, for example.
We also suggest and provide evidence in support of an explicit formula for the
entanglement entropy of free fermions in any dimension , as the size of a subsystem
, where is the Fermi surface and
is the boundary of the region in real space. The expression for the constant
is based on a conjecture due to H. Widom. We
prove that a similar expression holds for the particle number fluctuations and
use it to prove a two sided estimates on the entropy .Comment: Final versio
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Inelastic Effects in Low-Energy Electron Reflectivity of Two-dimensional Materials
A simple method is proposed for inclusion of inelastic effects (electron
absorption) in computations of low-energy electron reflectivity (LEER) spectra.
The theoretical spectra are formulated by matching of electron wavefunctions
obtained from first-principles computations in a repeated vacuum-slab-vacuum
geometry. Inelastic effects are included by allowing these states to decay in
time in accordance with an imaginary term in the potential of the slab, and by
mixing of the slab states in accordance with the same type of distribution as
occurs in a free-electron model. LEER spectra are computed for various
two-dimensional materials, including free-standing multilayer graphene,
graphene on copper substrates, and hexagonal boron nitride (h-BN) on cobalt
substrates.Comment: 21 pages, 7 figure
- …